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Mirrors > Home > MPE Home > Th. List > mircgrextend | Structured version Visualization version GIF version |
Description: Link congruence over a pair of mirror points. cf tgcgrextend 28328. (Contributed by Thierry Arnoux, 4-Oct-2020.) |
Ref | Expression |
---|---|
mirval.p | β’ π = (BaseβπΊ) |
mirval.d | β’ β = (distβπΊ) |
mirval.i | β’ πΌ = (ItvβπΊ) |
mirval.l | β’ πΏ = (LineGβπΊ) |
mirval.s | β’ π = (pInvGβπΊ) |
mirval.g | β’ (π β πΊ β TarskiG) |
mirtrcgr.e | β’ βΌ = (cgrGβπΊ) |
mirtrcgr.m | β’ π = (πβπ΅) |
mirtrcgr.n | β’ π = (πβπ) |
mirtrcgr.a | β’ (π β π΄ β π) |
mirtrcgr.b | β’ (π β π΅ β π) |
mirtrcgr.x | β’ (π β π β π) |
mirtrcgr.y | β’ (π β π β π) |
mircgrextend.1 | β’ (π β (π΄ β π΅) = (π β π)) |
Ref | Expression |
---|---|
mircgrextend | β’ (π β (π΄ β (πβπ΄)) = (π β (πβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . 2 β’ π = (BaseβπΊ) | |
2 | mirval.d | . 2 β’ β = (distβπΊ) | |
3 | mirval.i | . 2 β’ πΌ = (ItvβπΊ) | |
4 | mirval.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | mirtrcgr.a | . 2 β’ (π β π΄ β π) | |
6 | mirtrcgr.b | . 2 β’ (π β π΅ β π) | |
7 | mirval.l | . . 3 β’ πΏ = (LineGβπΊ) | |
8 | mirval.s | . . 3 β’ π = (pInvGβπΊ) | |
9 | mirtrcgr.m | . . 3 β’ π = (πβπ΅) | |
10 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircl 28504 | . 2 β’ (π β (πβπ΄) β π) |
11 | mirtrcgr.x | . 2 β’ (π β π β π) | |
12 | mirtrcgr.y | . 2 β’ (π β π β π) | |
13 | mirtrcgr.n | . . 3 β’ π = (πβπ) | |
14 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircl 28504 | . 2 β’ (π β (πβπ) β π) |
15 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mirbtwn 28501 | . . 3 β’ (π β π΅ β ((πβπ΄)πΌπ΄)) |
16 | 1, 2, 3, 4, 10, 6, 5, 15 | tgbtwncom 28331 | . 2 β’ (π β π΅ β (π΄πΌ(πβπ΄))) |
17 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mirbtwn 28501 | . . 3 β’ (π β π β ((πβπ)πΌπ)) |
18 | 1, 2, 3, 4, 14, 12, 11, 17 | tgbtwncom 28331 | . 2 β’ (π β π β (ππΌ(πβπ))) |
19 | mircgrextend.1 | . 2 β’ (π β (π΄ β π΅) = (π β π)) | |
20 | 1, 2, 3, 4, 5, 6, 11, 12, 19 | tgcgrcomlr 28323 | . . 3 β’ (π β (π΅ β π΄) = (π β π)) |
21 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircgr 28500 | . . 3 β’ (π β (π΅ β (πβπ΄)) = (π΅ β π΄)) |
22 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircgr 28500 | . . 3 β’ (π β (π β (πβπ)) = (π β π)) |
23 | 20, 21, 22 | 3eqtr4d 2775 | . 2 β’ (π β (π΅ β (πβπ΄)) = (π β (πβπ))) |
24 | 1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23 | tgcgrextend 28328 | 1 β’ (π β (π΄ β (πβπ΄)) = (π β (πβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6543 (class class class)co 7413 Basecbs 17174 distcds 17236 TarskiGcstrkg 28270 Itvcitv 28276 LineGclng 28277 cgrGccgrg 28353 pInvGcmir 28495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-trkgc 28291 df-trkgb 28292 df-trkgcb 28293 df-trkg 28296 df-mir 28496 |
This theorem is referenced by: mirtrcgr 28526 |
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